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Consider the following theory comprising of $n$ bosons and $n$ fermions (along with their conjugates) on a Riemannian Manifold, with arc length parameter $t$ (section 10.4.1, Mirror Symmetry by Vafa et al.):$$L=\frac{1}{2}g_{ij}\dot{\phi}^i\dot{\phi}^j+\frac{i}{2}g_{ij}\bigl(\bar{\psi}^iD_t\psi^j-D_t\bar{\psi}^i\psi^j\bigr)-\frac{1}{4}R_{ijkl}\psi^i\psi^j\bar{\psi}^k\bar{\psi}^l\tag{1}$$with the fermion covariant derivative:$$D_t\psi^i=\partial_t\psi^i+\dot{\phi}^j\Gamma^i_{jk}\psi^k\tag{2}$$I am having a problem deriving the following conjugate momenta for $\phi^i$ and $\psi^m$:$$p_m=\frac{\partial L}{\partial \dot{\phi}^m}=g_{mj}\dot{\phi}^j\tag{3}$$$$\pi_{\psi^m}=ig_{mj}\bar{\psi}^j\tag{4}$$ Here are the details of the issues that I'm having:

Problem: From the fact that the momentum conjugate to $\bar{\psi}^i$ has not been given, I deduce that it must be zero, so I try to use an integration by parts to absorb the two terms enclosed within the brackets in $(1)$ in to one single term. The result is the following Lagrangian:$$L=\frac{1}{2}g_{ij}\dot{\phi}^i\dot{\phi}^j+ig_{ij}\bar{\psi}^iD_t\psi^j-\frac{1}{4}R_{ijkl}\psi^i\psi^j\bar{\psi}^k\bar{\psi}^l\tag{5}$$ Inflicting the partial derivatives I get the following result:$$\pi_{\psi^m}=\frac{\partial L}{\partial \dot{\psi}^m}=ig_{mj}\bar{\psi}^j\tag{6}$$$$\pi_{\bar{\psi}^m}=\frac{\partial L}{\partial \dot{\bar{\psi}}^m}=0\tag{7}$$So far I have the fermionic momenta correct. The issue arises when I try to compute the bosonic momentum:$$p_m=\frac{\partial L}{\partial \dot{\phi}^m}=g_{mj}\dot{\phi}^j+i\Gamma_{jkm}\bar{\psi}^j\psi^k\tag{8}$$Clearly there is an additional non vanishing term which makes $(8)$ differ from $(3)$. Alternatively, should we (leaving aside the fermionic momenta for a while) work with $(1)$ thinking that two such terms may cancel, we get:$$p_m=\frac{\partial L}{\partial \dot{\phi}^m}=g_{mj}\dot{\phi}^j+\frac{i}{2}\bigl(\partial_jg_{km}-\partial_kg_{jm}\bigr)\bar{\psi}^k\psi^j\tag{9}$$Again this is non vanishing term. Is the fermionic covariant derivative to be treated independent of $\dot{\phi}^m$ so that it is killed by the derivative operator $\frac{\partial}{\partial \dot{\phi}^m}$? Or is $(3)$ a typing error? Or is it something else? Kindly help out.

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  • Yes, OP's eq. (8) for the canonical momentum is correct while eq. (3) is indeed an typo on the top of p. 208 in Ref. 1. See also point 6 in my Phys.SE answer here.

  • Calculating the fermionic canonical momentum is subtle for reasons mention in point 2 of the same answer.

References:

  1. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry, 2003. The PDF file is available here.
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